Langevins stochastic differential equation extended by a time-delayed term

The stochastic differential equation is a generalization of Lange-vin's equation, which is obtained if b = 0. Necessary and sufficient conditions on a, b and r are given under which a stationary solution exists. In this case, it is unique and Gaussian. Its covariance function and its spectral density are studied. The covariance function tends to zero exponentially, and the exponential rate of convergence is non-smooth for certain values of a, b and r Moreover, the covariance function oscillates around zero for certain parameters a, b, r